Randomly Weighted $d-$complexes: Minimal Spanning Acycles and Persistence Diagrams

Understand

A weighted $d-$complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight.

• We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes.
• First, we establish an equivalence between the MSA face-weights and \emph{death times} in the persistence diagram.
• Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately.